Doubling a square

In the Śulba-Sūtra, written around 800 BCE, Baudhāyana sets a problem: how can you build a square with double the area of a given square?

The first guess is to double each side. Try it and it fails badly: a square of side 2 has area 2 × 2 = 4 times the original, not 2 times. Doubling the side quadruples the area. (Almost everyone guesses this first.)

Baudhāyana's answer, in Verse 1.9, is elegant: 'The diagonal of a square produces a square of double the area of the original square.' So build the new square on the diagonal of the old one.

Why does that work? Baudhāyana's own constructions use what he calls east-west and north-south lines (horizontal and vertical lines at right angles to each other), and drawing a few of them makes it visible.

Extend the sides of the original square. They pass through the corners of the tilted new square, because a line that bisects an angle of a square passes through the opposite corner.

Now count triangles. The original square is made of two small triangles, split by its own diagonal. The new square on that diagonal is made of four of those same triangles.

And they're all congruent to one another (they fit exactly on top of each other, so they have equal areas), which is what makes the count a proof and not just a picture. Four triangles against two: exactly double the area.

Better still, the construction repeats. Take the diagonal of the new square and build on that, and you get a third square of 8 small triangles. Then 16, then 32. Each step doubles the area, for ever.